Quickly Compute Definite Integrals Using the Fundamental Theorem

By Mark Ryan

Here is a super-duper shortcut integration theorem that you’ll use for the rest of your natural born days — or at least till the end of your stint with calculus. This shortcut method is all you need for most integration word problems.

The fundamental theorem of calculus (second version or shortcut version): Let F be any antiderivative of the function f; then


This theorem gives you the super shortcut for computing a definite integral like


the area under the parabola y = x2 + 1 between 2 and 3. You can get this area by subtracting the area between 0 and 2 from the area between 0 and 3, but to do that you need to know that the particular area function sweeping out area beginning at zero,


(with a C value of zero).

The beauty of the shortcut theorem is that you don’t have to even use an area function like


You just find any antiderivative, F (x), of your function, and do the subtraction, F (b) – F (a). The simplest antiderivative to use is the one where C = 0. So here’s how you use the theorem to find the area under your parabola from 2 to 3.


is an antiderivative of x2 + 1. Then the theorem gives you:


Regardless of the function, this shortcut works, and you don’t have to worry about area functions. All you do is F (b) – F (a).

Here’s another example: What’s the area under f (x) = ex, between x = 3 and x = 5? The derivative of ex is ex, so ex is an antiderivative of ex, and thus


What could be simpler?

Areas above the curve and below the x-axis count as negative areas. Before going on, it’s important to touch on negative areas. Note that with the two examples shown here, the parabola, y = x2 + 1, and the exponential function, y = ex, the areas you’re computing are under the curves and above the x-axis. These areas count as ordinary, positive areas. But, if a function goes below the x-axis, areas above the curve and below the x-axis count as negative areas.

Okay, so now you’ve got the super shortcut for computing the area under a curve. And if one big shortcut wasn’t enough to make your day, This table lists some rules about definite integrals that can make your life much easier.